On commuting matrices and pencils
(2026) Linear Algebra and its Applications — Vol. 738, p. 25-44 (2026)
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- Authors
Van Dooren, Paul 
- Abstract
- In this paper we extend the classical notion of commuting matrices to that of commuting pencils. This definition requires the pencils to be regular but they may have singular matrices. If the matrices happen to be invertible, our extension is equivalent to the commutativity of the (left) quotients . This choice is linked to the application of commuting sets of descriptor systems [1]. We show that the invertibility condition of the matrices can be circumvented by using Möbius transforms of the pencils . We also introduce minimization problems that can serve as necessary and sufficient condition for the commutativity of certain classes of diagonalizable pencils. The more challenging non-diagonalizable case is not considered here.
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Citations
Benner, P., & Van Dooren, P. (2026). On commuting matrices and pencils. Linear Algebra and its Applications, 738, 25-44. https://doi.org/10.1016/j.laa.2026.02.032 (Original work published 2026)